Impossibility of Collusion under Imperfect Monitoring with Flexible Production.

Summary:

We show that it is impossible to achieve collusion in a duopoly when (a) goods are homogenous and firms compete in quantities; (b) new, noisy information arrives continuously, without sudden events; and (c) firms are able to respond to new information quickly. The result holds even if we allow for asymmetric equilibria or monetary transfers. The intuition is that the flexibility to respond quickly to new information unravels any collusive scheme. Our result applies to both a simple stationary model and a more complicated one, with prices following a mean-reverting Markov process, as well as to models of dynamic cooperation in many other settings. (JEL D43, L12, L13)ABSTRACT FROM AUTHORCopyright of American Economic Review is the property of American Economic Association and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.

Excerpt from Article:

1794 Collusion is a major problem in many mar- kets and has been an important topic of study in both applied and theoretical economics. From exposed collusive cases, we know how numer- ous real-life cartels have been organized and what kinds of agreements (either implicit or explicit) are likely to be successful in obtain- ing collusive pr;ts.1 At least since George J. Stigler (1964), economists have recognized that imperfect monitoring may destabilize cartels. Nevertheless, the seminal paper of Edward J. Green and Robert H. Porter (1984) has shown that, even with imperfect monitoring,;rms can create collusive incentives by allowing price wars to break out with positive probability. 1 There are many papers describing explicit and tacit collusion among;rms. For comprehensive studies, see, for example, George A. Hay and Daniel Kelley (1974), Margaret C. Levenstein and Valerie Y. Suslow (2006), or Joseph E. Harrington (2006). Impossibility of Collusion under Imperfect Monitoring with Flexible Production By ; ;; ; ;* We show that it is impossible to achieve collusion in a duopoly when (a) goods are homogenous and;rms compete in quantities; (b) new, noisy information arrives continuously, without sudden events; and (c);rms are able to respond to new information quickly. The result holds even if we allow for asymmetric equilibria or monetary transfers. The intuition is that the;exibility to respond quickly to new information unravels any collusive scheme. Our result applies to both a simple sta- tionary model and a more complicated one, with prices following a mean-reverting Markov process, as well as to models of dynamic cooperation in many other set- tings. (JEL D43, L12, L13) We study the scope of collusion in a quan- tity-setting duopoly with homogenous goods and;exible production—that is, if;rms can change output;ow frequently.2 As in Green and Porter (1984), in our model;rms cannot observe each other’s production decisions directly. They observe only noisy market prices/signals that depend on the total market supply. We show that collusion is impossible to achieve if: 1. New, noisy information arrives continu- ously, without sudden events; 2. The;rms have;exible production technol- ogies and can thus react to new information quickly; and 3. Public signals depend on total market sup- ply only, and not on individual decisions. There are many markets that;t the general features of our model. For example, in markets with homogenous goods, e.g., chemicals,;rms are selling both to a spot market and to clients, with client deals being private but affecting the 2 The term;exible production is usually understood as describing low costs of changing the amount and the variety of output. In this paper, we use this term in a nar- rower sense, having;rms face low costs of changing;ow but restricting them to produce only one type of output. Flexibility on the variety dimension has separate effects on the scope of collusion: increased product differentiation may improve collusion, but increased complexity of moni- toring may destabilize cartels. * Sannikov: Department of Economics, 549 Evans Hall, University of California, Berkeley, CA 94720 (e-mail: sannikov@econ.berkeley.edu); Skrzypacz: Stanford Grad- uate School of Business, 518 Memorial Way, Stanford, CA 94305 (e-mail: andy@gsb.stanford.edu). We thank Jeremy Bulow, Kyna Fong, Drew Fudenberg, Joseph Harrington, Ichiro Obara, Yuval Salant, Robert Wilson, and seminar participants at the SED 2004, IIOC 2005, Midwest Macro 2005, SITE 2005, Arizona State University, UC Berkeley, Columbia University, University of Chicago, University of Illinois in Urbana Champaign, University of Iowa, New York University, Penn State University, University of Pennsylvania, Princeton University, Rutgers Univer- sity, Stanford University, University of Texas in Austin, and Washington University for useful comments and suggestions.



VOL. 97 NO. 5 1795SANNIKOV AND SKRZYPACZ: IMPOSSIBILITY OF COLLUSION spot market. The spot price can be used to moni- tor the success of collusion. One example that received a lot of attention was the cartel produc- ing lysine, an amino acid.3 This cartel tried to collude by setting and monitoring a target price at;rst. However, those early attempts failed. Quoting from Cabral (2005, 201): The topic of lysine prices came up at a dinner meeting in Chicago between ADM and European executives. The latter com- plained about low prices and accused ADM of being responsible for it. ADM’s Whitacre responded that “one can point a lot of;ngers,” and that the best thing to do was to;nd a solution to the problem. The cartel has encountered the di;culty related to condition 3: they could not identify the deviator.4 What solution did they agree upon? Along the lines of Green and Porter (1984), they could have agreed upon a new target price, com- mitting to go to a price war if the price fell below the new target. However, that would repeat the old story: as our results suggest, that would not have worked. Instead, the solution was to let output;gures “be collected every month by the trade association … . If one company sold more than it was allotted, it would be forced to pur- chase lysine from companies lagging behind” (Eichenwald 2000, 205). Thus, the cartel began 3 The lysine cartel has received a lot of attention thanks to the abundance of detailed information available on its inner workings, including FBI videotapes of cartel mem- bers’ meetings. The cartel was described in detail by Kurt Eichenwald (2000) in a 600-page book, and discussed fur- ther by Luis M. B. Cabral (2005) and Harrington (2006). 4 The problems were real—ADM was indeed over- producing. collecting individual company data and aggre- gating that data over monthly periods, breaking conditions 1 and 3. The practice of collecting data on market shares has been especially common among car- tels.5 Even the Joint Executive Committee rail- road cartel, the motivation for the Green and Porter (1984) model of equilibrium price wars,6 collected data on individual members’ market shares. Other cartels have also limited the;exi- bility of its members to respond to new informa- tion by setting strict rules regarding acceptable forms of contracts with customers and by col- lecting data about suspected deviations through secret investigations (e.g., see the sugar trust cartel described in David Genesove and Wallace P. Mullin 2001). The failure of the lysine cartel to collude by setting a target price at the beginning of its oper- ation illustrates how the provision of incentives can break down under;exible production, even when;rms have very clear information about the success of collusion. Figure 1 illustrates this sur- prising fact using a theoretical model presented in Section VI. In this example, spot prices are correlated over time and have an unconditional mean of 1 2 Q, where Q is the total quantity. Figure 1 shows two sample paths of prices when the;rms produce Cournot Nash quantities and when they split the monopoly quantity.7 Just by 5 For a comprehensive list of such arrangements, see Harrington (2006). 6 See Thomas Ulen (1983) and Porter (1983) for a detailed study of this cartel. 7 In this nonstationary setting, the absence of collusion is characterized by a Markov perfect equilibrium (MPE), and the;rst-best collusion is similarly a state-dependent strategy. However, the Nash equilibrium and monopoly quantity of a one-shot analogue of this game (with constant demand 1 – Q) serve well for illustration. ? ? ? ? ? ? ? ? ? ? ?? ? ?? ?? ?ime P ric e Cournot supply Monopoly supply ; 1



DECEMBER 20071796 THE AMERICAN ECONOMIC REVIEW looking at the price level at any moment of time, it is obvious whether the;rms are colluding or not. Yet, despite this apparent transparency, collusion is impossible when;rms see prices continuously and act su;ciently frequently, and when prices depend only on the total supply. Why? A;rst guess may be that fast arrival of information and the;exibility to respond to it facilitate collusion, as;rms can punish poten- tial deviators more quickly. However, although this intuition is true in games with perfect mon- itoring, it does not always hold in games with imperfect monitoring, as demonstrated for the ;rst time in the classic paper of Dilip Abreu, Paul Milgrom, and David Pearce (1991) (here- after AMP).8 Let us see why collusion is impos- sible under the wide range of conditions we study. First, let us consider the classic case of a stationary repeated game with a strongly sym- metric collusive scheme, in which;rms behave identically after all histories. Following Abreu, Pearce, and Ennio Stacchetti (1986) (hereaf- ter APS’86), an optimal symmetric equilib- rium has two regimes: a collusive regime and a price war–punishment regime. In the collusive regime,;rms produce less than the static Nash equilibrium quantities. If the price drops below a critical level, this arouses enough suspicion of cheating to cause a price war. In the price war regime,;rms strongly overproduce, because the intensity of the price war (i.e., low prices) makes the return to the collusive regime more likely. With shorter time periods between actions, ;rms must decide whether to trigger a price war by looking at noisier incremental information. As we show, this causes;rms to make type I errors by triggering price wars on the equilib- rium path disproportionately often, erasing all ben;ts from collusion. To see this intuitively, let us compare games in which the time period between actions is either D or 2D, where D is small. Suppose that information arrives contin- uously, so that the aggregate summary statistic of the information in each period is normally distributed. In Figure 2, the horizontal axis illustrates the summary statistic (e.g., average 8 AMP pioneered the theory of games with frequent actions. We discuss the relationship to their paper at the end of the introduction, and, in more detail, at the end of Section IIA. price) for one period of length D, and the verti- cal axis illustrates the summary statistic in the next period of length D. In an optimal symmet- ric equilibrium, at the end of each period;rms test the summary statistic against a cutoff level to decide whether to trigger a price war. Figure 2 illustrates the critical regions 1 and 2 that trig- ger a price war in a game with period length D. If the time period between moves increases to 2D, two forces i;uence the scope of collusion. First,;rms learn more information per period, which helps collusion. Second, the gain from deviation in a given period increases, which hurts collusion.9 From Figure 2, we learn that the;rst effect is stronger when D is small, i.e., collusion is more di;cult with smaller time periods between moves. Indeed, note that at the end of a period of length 2D we can perform the same tests to decide about a price war as in a game with periods of length D, and these tests provide approximately the right incentives against a deviation with periods 2D.10 However, if;rms can change actions only once per time interval 2D, a more effective joint test (illus- trated by the dashed diagonal line) can provide the same incentives more e;ciently. Therefore, collusion is more e;cient for time periods of length 2D.11 As we show in this paper, with infor- 9 Also, punishment is delayed when;rms keep quanti- ties;xed for a period of time of 2D, which hurts collusion, but this effect is negligible for small D. 10 Strictly speaking, if we perform tests with criti- cal regions 1 and 2 in a game with period 2D, they pro- vide a bit weaker incentives than in a game with period D, because when the;rst signal falls in region 1, punishment is delayed by time D. However, for small D this difference is negligible. 11 We know from APS’86 that the most e;cient equilib- rium does not use review strategies, and so the;rms can- not use the more e;cient two-period tests when the period Price in period 1 Price in period 2 Region 1 Region 2 ; 2



VOL. 97 NO. 5 1797SANNIKOV AND SKRZYPACZ: IMPOSSIBILITY OF COLLUSION mation arriving continuously, the e;ciency loss for small D is so large that collusion becomes impossible altogether. In Section III we show that collusion is not possible in asymmetric equilibria, even if play- ers use monetary transfers. In many games players can enforce collusive schemes without price wars by using asymmetric continuation equilibria or monetary transfers. Namely, rather than destroying value, they can transfer payoffs among themselves, keeping the average pr;ts high. As we show, our environment causes such collusive schemes to fail, a result that involves new insights. The main intuition is that with a one-dimensional signal and a continuum of quantities to choose from, transfers used to pro- vide incentives for one player interfere with the incentives of the other player. As a result, collu- sion cannot be sustained by transfers alone. The provision of incentives necessarily involves the destruction of value.12 Figure 3 shows a stylized relationship between the length of the period D and the scope of col- lusion (see Figure 6 in Section IV for a detailed example). As we see, the relationship is not monotonic and the highest collusive payoffs are achieved at an interior value of D. Moreover, as D becomes small, the highest equilibrium prof- its decrease to the stage-game Nash equilibrium pr;ts. Besides looking at a classic stationary model of Cournot duopoly with constant marginal costs, we explore a number of extensions of applied interest. In Section V we consider increasing marginal costs and capacity constraints. In between actions is D. Indeed, incentives would break down if the;rms tried to do that. First, after seeing the;rst-period prices, the;rms will likely react, either by reducing pro- duction to avoid a price war, or by increasing production if the;rst-period outcome makes it unlikely that the price war will be triggered after the second-period review. Second, anticipating that (own) reaction, each;rm will overproduce in the;rst period. Third, anticipating the reaction of the other;rm, a;rm expects its opponent to share the cost of preventing punishment in the second period and hence will overproduce in the;rst period even more. Since a single decision maker ben;ts from more;exibility (which is cap- tured by the;rst two effects), it is the third, strategic effect that prevents collusion as D S 0. 12 This intuition is related to the concept of ident;ability from Drew Fudenberg, David K. Levine, and Eric Maskin (1994); however, lack of ident;ability is not su;cient to establish the result. Section VI we extend the results to a more real- istic nonstationary model with correlated prices. In Section VII we show that even if the moni- toring technology is richer (so that deviations of the players affect signals differently), if there is a high correlation between the signals used to monitor the two players, the scope of collu- sion is substantially limited. Our results also extend to settings with more than two;rms and multiple signals. Finally, the end of Section VII shows that our result holds even if we allow for a class of private strategies, which were shown in Michihiro Kandori and Ichiro Obara (2006) to greatly improve the players’ payoffs in other environments. Our analysis of cartels extends to coopera- tion in many other types of dynamic interac- tion, including, for example, moral hazard in teams. In particular, we present a model of a partnership in which efforts of team members are private while publicly observed outcomes depend (stochastically) on the sum of efforts of the partners. In that environment, cooperation over and above static Nash equilibrium is not possible (as long as the marginal costs of effort are not too concave) if the partners observe the outcomes continuously and can react to infor- mation quickly.13 AMP have been the;rst to show that fre- quent actions can reduce the scope of collusion. They study symmetric equilibria in a prison- er’s dilemma, in which just one type of signal arrives at a Poisson rate. This signal can be of 13 An example of such a partnership game is found in Roy Radner, Roger Myerson, and Maskin (1986), discussed below. Half monopoly profit Static Nash profit C Profit Best collusive payoff ; 3



DECEMBER 20071798 THE AMERICAN ECONOMIC REVIEW a good type, indicative of cooperation, or a bad type, indicative of defection. When players act frequently, it is very unlikely that more than one signal arrives in a given period: effectively, players observe at most one signal per period. If the signal is good, it cannot be used to trigger punishment, and cooperation becomes impos- sible with frequent actions. If the signal is bad, it can be used to trigger punishment, and limited cooperation is possible. The nature of the signals is one of the fun- damental differences between their model and ours. One is tempted to think that fre- quent actions should not hurt cooperation with Brownian signals. Tails of normally distributed signals are so informative about the players’ actions that perfect collusion should be possible if players are patient. Nevertheless, as D S 0 (i.e., as players become more patient and per- period information deteriorates), we show that, surprisingly, collusion is never possible (unlike with Poisson signals).14 Additionally, in comparison to AMP, we explore a number of other types of equilibria and many applied extensions. For example, we consider both symmetric and asymmetric equi- libria, each possibly with monetary transfers. In fact, even though AMP do not explore this issue, in their setting, asymmetric equilibria may achieve some cooperation even when no cooperation is possible in symmetric equilib- ria. We show that collusion becomes impossible even when one considers a nonstationary set- ting and compares collusive payoffs with those in Markov perfect equilibria. We also explore the issues of increasing marginal costs, multiple signals, mixed strategies, and even equilibria in private strategies.15 Another closely related paper is Radner, Myerson, and Maskin (1986). They study a repeated partnership game in which in every period there are only two possible outcomes: a success or a failure. The probability of success depends on the sum of efforts. They show that the best equilibria in their setting are uniformly 14 The differences between Brownian and Poisson infor- mation have been recently further explored by Fudenberg and Levine (2007) and Sannikov and Skrzypacz (2007). 15 We also analyze a different game—the Green and Porter duopoly with a continuum of action choices per period. bounded away from e;ciency for all discount rates. The intuition roughly consists of two parts:;rst, as in our paper, because it is di;cult to distinguish between deviations of different players, asymmetric equilibria do not improve upon symmetric ones. Second, because the sig- nals have bounded likelihood ratios, symmetric equilibria are necessarily costly. In contrast, in our setup the likelihood ratios are unbounded, so taking the interest rate to zero (while keep- ing D;xed) leads to asymptotically;rst-best cooperation. If, however, we take D to zero, no cooperation is possible. There is a large and growing theoretical liter- ature trying to assess the impact of information on the scope of collusion in an environment with imperfect monitoring. Green and Porter (1984) were the;rst to propose a symmetric collusive equilibrium in which price wars are used on the equilibrium path to prevent deviations. APS’86 characterize optimal symmetric equilibria in this setting and show that they involve two extreme regimes. The general analysis of games with hidden actions has been extended to asymmet- ric equilibria by Abreu, Pearce, and Stacchetti (1990), Fudenberg, Levine, and Maskin (1994), and Sannikov (2007), among others.16 The paper is organized as follows. Section I presents a simple model of a repeated game with stationary prices. Section II proves the main result for symmetric public perfect equilibria. Section III proves the result for asymmetric equilibria with and without monetary trans- 16 Many papers look at the role of asymmetric equilibria and monetary transfers in achieving collusion. In a static set- ting with private information, R. Preston McAfee and John McMillan (1992) show the necessity of transfers for any collusive scheme utilizing the private information. Susan Athey and Kyle Bagwell (2001), Masaki Aoyagi (2003), Andreas Blume and Paul Heidhues (2001), and Skrzypacz and Hugo Hopenhayn (2004), among others, extend that intuition by studying collusion in repeated games without monetary transfers and emphasizing the need for transfers of continuation payoffs. Harrington and Skrzypacz (2007) study repeated competition with hidden prices and observed stochastic market shares, and show that symmetric equi- libria (with price wars after skewed market shares) do not improve upon competitive outcomes, but asymmetric equi- libria do. Similarly, Athey and Bagwell (2001) show that asymmetric strategies greatly improve the scope of collu- sion (which is shown to be quite limited in the symmetric equilibria of Athey, Bagwell, and Chris Sanchirico (2004)). In contrast, our impossibility result holds even for asym- metric equilibria and with any monetary transfers.



VOL. 97 NO. 5 1799SANNIKOV AND SKRZYPACZ: IMPOSSIBILITY OF COLLUSION fers. Section IV presents a numerical example. Section V explores the possibility of nonlinear costs and capacity constraints. Section VI proves our impossibility result in a more complicated and realistic model in which market prices are correlated over time. Section VII discusses vari- ous mod;cations of the model. Section VIII concludes. I. The Stationary Model Two;rms compete in a stationary market with homogenous products. The time horizon is i;nite and;rms discount future pr;ts with a common interest rate r. Firms set quantities every period and the resulting prices depend on quantities and noise. We study how the scope of potential collusion depends on the;rms’;ex- ibility of production. We describe the;exibility of production in terms of D, the duration of time periods between which production decisions are made. In a repeated game GD,;rms play the stage game at time points t 5 0, D, 2D, … . The stage game is as follows: at time t 5 nD the;rms choose (privately) supply rates qit [ 30, q¯4 for a time interval 3t, t 1 D2 (where q¯ is a large, exogenous capacity constraint). Denote by Qt 5 q1t 1 q2t the total supply rate of the two;rms. During this time interval 3t, t 1 D2;rms supply DQt of the product. Both;rms have the same constant marginal cost, which we normalize to zero.17 The pr;t of each;rm is just the revenue, which depends on the price and the supply in a given period. Prices are publicly observable and depend on the total supply as well as a random shock pt 5 P 1Qt 2 1 et , where the inverse demand function P: 30, 2q¯4 S R is strictly decreasing, twice continuously dif- ferentiable, and P 102 . 0. Firm i’s revenue in time interval 3t, t 1 D2 is Dqit 1P 1Qt 2 1 et 2 . 17 See Section V for the extension to increasing mar- ginal cost. To facilitate comparisons of equilibrium outcomes for different D, assume that random shocks come from a standard Brownian motion 5Zt ; t $ 06. Shocks et are formed in a way that makes total revenue of each;rm depend not on D, but only on the supply rates. Spec;cally, the revenue that;rm i gets from time 0 to t is a s50, D, 2D...t2D Dqis 1P 1q1s 1 q2s 2 1 es 2 5 3 t 0 qis 1P 1q1s 1 q2s 2 ds 1 sdZs 2 , where qis with s [ 3nD, nD 1 D2 is the supply rate that;rm i chooses at time nD, and s2 is the variance of the noise. Then et , the average price shock over time interval 3t, t 1 D2 , is simply et 5 s D 1Zt1D 2 Zt 2 . Therefore, et , N 10, s2/D2 . Note that random shocks have greater variance over small time intervals, so that the information that;rms learn by observing prices is proportional to the time interval. We can also interpret the noise struc- ture in the following way. Suppose that market prices are quoted every second and that these prices have a normal distribution with mean P(Q) and variance s2. If;rms can change their supply rates every D seconds, then the average price over that interval has mean P(Q) and vari- ance s2/D (and as;rms are risk neutral, these two interpretations of pt are equivalent). The fact that average prices can become unboundedly negative due to very high vari- ance over short time periods is unattractive. However, it allows for a simple model in which the argument behind our main result is par- ticularly clean.18 With the simple model, we develop intuition that translates easily to more complex situations. See Section VI for a more realistic nonstationary model, with bounded instantaneous prices. Also, Section VII explains how our results extend to a stationary model in which the distribution of per-period prices is not 18 It is important to point out that our results do not fol- low simply from the per-period variance increasing to i;n- ity as D S 0, but rather from the rate at which it increases.



DECEMBER 20071800 THE AMERICAN ECONOMIC REVIEW necessarily normal. In particular, prices can be a nonnegative function of total supply and ran- dom shock, and, for example, follow a lognor- mal distribution. A crucial assumption is that as in Green and Porter (1984) and Radner, Myerson, and Maskin (1986), the price depends on the total supply only, so that the distribution of prices changes in the same way regardless of whether;rm 1 or;rm 2 increases its production by a unit, no matter what production levels they start with. That is, we assume that the products are homog- enous. As demonstrated in Sections VI and VII, the assumption that the random price shocks are distributed normally and independently of past prices is not essential for the results. In the repeated game,;rms choose sup- ply rates after every history to maximize their expected discounted pr;t. Firm i’s (normal- ized) expected payoff is E c 11 2 e2rD2 a t50, D, 2D... e2rtqit 1P 1Qt 2 1 et 2 d , where the expectation takes into account the dependence of future supply rates on past prices. A history of;rm i at time t is the sequence of price realizations and own supply decisions up to time t. A public history contains only the realizations of the past prices.19 We analyze the pure strategy public perfect equilibria (PPE) of the game. A strategy of a;rm is public if it depends only on the public history of the game. Two public strategies form a PPE if after any public history the continuation strategies form a Nash equilibrium. Considering only pure strat- egies is admittedly restrictive. In Section VII we argue that the intuition holds also for public mixed strategies and for an important class of private mixed strategies that have been shown to improve collusive payoffs in other settings.20 19 One can also allow for public randomizations, so that before each period the players observe a realization of a public random variable (which becomes a part of the public history). It does not change any of the results: since prices have full support, any randomization can be supported using only prices. 20 When we consider pure strategies, it is not restrictive to focus on public strategies. Indeed, for every private strat- egy, one can;nd a public strategy that induces the same probability measure over private histories. Throughout the paper, we use the term PPE in the sense of pure strategy perfect public equilibria. Remark 1: In the abstract and the introduction we have used extensively an informal phrase, “information arrives continuously.” We can now explain this term formally. If prices are i.i.d. and the average price from time 0 to time t is a continuous function of t, then prices must take the form spec;ed in our model (in particular, the noise has to be generated by a Brownian motion). Formally speaking, if P¯t, the average price between times 0 and t, is continuous, then tP¯t is a Lévy process without jumps. Such a pro- cess can be represented as a sum of drift and diffusion terms, so average prices in each period are normally distributed. Remark 2: It is important to note that the;ex- ibility D plays two roles in our model: it affects the variance of the price distribution in a given period, and it affects the per-period discount rate d K e2rD. The;rst effect makes collusion more di;cult for smaller D since it results in less precise statistical inference, which makes deviations more di;cult to detect. The second effect (standard in repeated games) makes it easier to collude because the single-period ben- ;ts to deviation become small relative to the continuation payoffs. Remark 3: The methods and results for the repeated duopoly game are applicable far beyond the realm of collusion. To give an illustration, let us present a stylized model of moral hazard in teams, in which payoffs above the static Nash equilibrium are not achievable as D S 0 for the same reasons as in a repeated duopoly setting. Two partners choose effort rates 1q1t, q2t 2 that affect the partnership’s pr;t. The average pr;t per period (gross of private costs of effort) is Rt 5 P 1q1t 1 q2t 2 1 et, where P9 1 · 2 . 0 and et , N 10, s2/D2 . The pay- offs to partner i are 11 2 e2rD2 a t50, D, 2D... e2rt ARit 2 c 1qit 2 B ,



VOL. 97 NO. 5 1801SANNIKOV AND SKRZYPACZ: IMPOSSIBILITY OF COLLUSION where Rit is the share of pr;t;ow that player i receives at time t and c 1q 2 is the cost of effort (increasing and convex). Effort by player i ben- ;ts both players, but only player i pays its cost. This externality implies that the static Nash equilibrium does not yield joint pr;t maximi- zation. Even though there is no direct mapping between this model and the duopoly model we study, the assumptions that information arrives continuously and that total pr;t depends only on joint effort drive the result that, as D S 0, the highest payoffs the partnership can achieve are the static Nash equilibrium payoffs.21 The stage game is similar to Radner, Myerson, and Maskin (1986), with the main differences being that we have normally distributed pr;ts while they have a binary distribution, and that we take D S 0 while they take r S 0. A. Structure of the Stage Game We make the following two assumptions about the inverse demand function: • A1: The marginal revenue (of total demand) is decreasing: 102/0Q22 1QP 1Q2 2 , 0. • A2: The static best response to any q [ 30, q¯4 is less than q¯, i.e., the marginal revenue of residual demand is negative at q¯: 30 1q9P 1q9 1 q 2 2/0q94 Z q9 5 q¯ , 0. Assumptions A1 and A2 are su;cient to guar- antee that the best response in the stage game, q* 1q 2 , is unique and that the Nash equilibrium is symmetric and unique. Denote the Nash equi- librium of the stage game by 1qN, qN2 . We refer to it also as the “competitive equilibrium” or the “static equilibrium,” especially when we talk about the repetition of 1qN, qN2 in each stage of the dynamic game. D;ne vN 5 P 12qN2qN. LEMMA 1: Assume A1 and A2. The static best response q* 1q 2 is unique and less than q¯. Also, the static Nash equilibrium is symmetric and unique. 21 This result holds under assumptions similar to the ones we make in the duopoly model. See Remark in Section VB (p. 1808) for further discussion. The proof is standard and can be found in the Appendix. II. Impossibility of Collusion: Symmetric Equilibria In this section, we prove that collusion be- comes impossible as D S 0 in symmetric PPE. From APS’86, we know that optimal symmet- ric PPE involve two regimes: a collusive regime and a price war regime. In each period, the deci- sion whether to remain in the same regime or to switch is guided by the outcome of the price in that period alone. We show that as D S 0, tran- sitions from the collusive regime to the price war regime must happen very frequently to provide the players with incentives. Because of that, price wars destroy all collusive gains, and payoffs above static Nash become impossible in the limit. The methods we develop in this sec- tion are important for asymmetric PPE (possibly with monetary transfers) as well, but the analysis of those will require further insights. Denote by 3v ¯ 1D2 , v¯ 1D2 4 , R the set of payoffs achievable in symmetric PPE in the game GD. We would like to show that as D S 0, v¯ 1D2 converges to the “com- petitive payoff”, vN 5 P 12qN2qN. First, let us informally present the core of our argument. Using the results of APS’86, in the collusive regime, players expect total pay- offs v¯ 1D2 and choose some supply rates 1q, q 2 , 1qN, qN2 in the current period. In the next period, depending on the realized price, players either stay in the collusive regime with continuation payoff v¯ 1D2 or go to a price war with continu- ation payoff v ¯ 1D2 . The lower the punishment v ¯ 1D2 is, the higher the collusive payoffs v¯ 1D2 can be: harsher punishments make providing incen- tives easier. We will show that v¯ 1D2 converges to vN even if players can use a punishment of 0 (clearly v ¯ 1D2 $ 0 because 0 is the minmax payoff—a;rm can guarantee itself that payoff by producing 0). As D S 0, collusion becomes impossible because the statistical test to prevent deviations sends players to a price war with a disproportionately high probability. To see this in greater detail, consider a devia- tion from q to a static best response q* 1q 2 . q that reduces the mean of the observed average price from m 5 P 12q 2 to m9 5 P 1q* 1q 2 1 q 2 , m. Lemma 3 below shows that the best statis- tical test to prevent this deviation is a tail test,



DECEMBER 20071802 THE AMERICAN ECONOMIC REVIEW which triggers a punishment when the price falls within a critical region 12 ,` c 4 . Given such a test, a deviation increases the probability of punishment by likelihood difference 5 G9 1c 2 2 G 1c 2 , where G9 and G are normal cumulative distri- bution functions with variance s2/D and means m9 and m, respectively. The probability of type I error, i.e., triggering punishment when no devi- ation has occurred, is given by the size of the test: size 5 G 1c 2 . Because the gain from a deviation in one period is on the order of D, we explore tail tests with a likelihood difference on the order of D. Lemma 2 shows that in such tests, as D S 0, the prob- ability of making a type I error in each period blows up relative to D. Therefore, as illustrated in Figure 4, v¯ 1D2 cannot be sustained. The rea- soning is as follows. The total expected payoff is a weighted sum of the current period payoff and the expected continuation payoff with weights equal to 11 2 d 2 and d. The continuation pay- off is a weighted average of v¯ 1D2 and 0 (recall that we allow for punishments that are at least as harsh as the equilibrium ones, which relaxes the problem) with weights 1 2 G 1c 2 and G 1c 2 , respectively: v¯ 1D2 5 11 2 d 2qP 12q 2 1 d Cv¯ 1D2 A1 2 G 1c 2 B 1 0 · G 1c 2 D . This is represented graphically as Figure 4. Now, as D S 0, we have two effects: the pay- off gain from colluding in the current period 11 2 d 2 AqP 12q 2 2 qNP 12qN2 B becomes small, and the payoff loss due to punishment becomes small. However, the payoff loss of dv¯ 1D2G 1c 2 becomes disproportionately large compared to the current-period payoff gain in the limit. As a result, any payoff above Nash cannot be sus- tained, even if we both: • Worry about only one deviation to a static best response; and • Allow 0 as the harshest punishment, even though typically v ¯ 1D2 . 0. When comparing games with different D, one may be tempted to employ the following reason- ing. Consider two games with respective time periods D and D9 5 D/2. Suppose that with D we can construct a pr;table collusive equilib- rium. Now, moving to D9, instead of employing the factorization techniques of APS’86 (namely, that it is su;cient to factorize the game into current period and continuation payoffs), con- sider the following strategies as a candidate for equilibrium. In the odd periods,;rms are rec- ommended to follow the same strategies as in the game with D, using the average price over the last two periods to decide on continuation play. In the even periods, the recommendation is to ignore the current prices and keep the quanti- ties;xed from last period. Clearly, if;rms follow these recommenda- tions, they will obtain the same payoffs as in the game with D. Unfortunately, these recommen- dations are not incentive compatible for several reasons. First, even if;rm 2 follows the recom- mendation,;rm 1, after seeing a high;rst-period price, has incentives to increase quantity in the second period, and after seeing a low price, to decrease quantity. That “option value” will also make;rm 1 increase its quantity in the;rst period. Second, by that same reasoning,;rm 1 will expect;rm 2 to react in the second period to the observed;rst-period price. That makes ;rm 1 increase its;rst-period quantity even more:;rm 1 gets all the ben;t, while the cost of reducing quantities in the second period if prices turn out to be low will be shared between the two;rms.22 22 Although the second of these two effects is the main reason cooperation is impossible for small D, clearly the two effects are related. 4 0vz *C+ vz *C+ 11G *c+ G *c+�-�O *C1/20d+ c *11c+4O *C+qP*2q+�-�vz *C+ ; 4



VOL. 97 NO. 5 1803SANNIKOV AND SKRZYPACZ: IMPOSSIBILITY OF COLLUSION A. Formal Argument First, we prove that as D S 0, the probability of type I error under a tail test to prevent devia- tions becomes disproportionately high. LEMMA 2: Fix C1 . 0 and m 2 m9 . 0. If D . 0 is su;ciently small, then a tail test for a deviation with a likelihood difference of C1D has a probability of type I error greater than O 1D0.51e 2 for any e . 0. PROOF: See Appendix. Next, we want to show that a tail test is an opti- mal way to deter any given deviation. Consider a deviation with instantaneous gain DD, which reduces the mean of observed prices from m to m9. The optimal test maximizes the expected continuation payoff subject to providing incen- tives against this deviation. Lemma 3 shows that if we are worried about only one deviation, a tail test with a bang-bang property is best. LEMMA 3: Suppose D . 0. Consider the problem max v1x2 3 ` 2` v 1x 2g 1x 2 dx s.t DD # d3 ` 2` v 1x 2 1g 1x 2 2 g9 1x 2 2 dx and 5x [ R, v 1x 2 [ 3v ¯ , v¯ 4 , where g and g9 are the densities of normal dis- tributions with variance ;/D and means m and m9 , m, respectively. If this problem has a solu- tion, then it takes the form v¯ if x . c(1) v 1x 2 5 e v ¯ if x # c for some c [ 12`, 1m 1 m92/24 . PROOF: See Appendix. With the help of Lemmas 2 and 3 we are ready to formulate our main result. PROPOSITION 1: As the time period between actions converges to zero, the maximal payoff achievable in symmetric PPE converges to the static Nash equilibrium payoff, i.e., v¯ 1D2 S vN as D S 0. PROOF: See Appendix. As D S 0, two effects i;uence the scope of collusion: more frequent moves lead to both weaker statistical tests (which makes collusion more di;cult to sustain) and smaller immedi- ate ben;ts to deviation (which makes collusion easier to sustain). The proof shows that as D S 0, the deterioration of information is so extreme that collusion becomes impossible altogether. We;nish this section by comparing the results to AMP, where information arrives discontinu- ously via Poisson jumps. Suppose that instead of continuous monitoring via prices, the players can monitor deviations through public signals that arrive according to a Poisson arrival rate, which depends on the total supply Q. To facili- tate comparison, focus on a single deviation to a static best response. This makes the setup very similar to the repeated prisoner’s dilemma game in AMP. There are two cases to consider with such discontinuous monitoring: the bad news case (arrival rate increasing in Q, so that arrival rate is higher upon deviation) and the good news case (arrival rate decreasing in Q). In the best collusive symmetric equilibria, incentives are provided quite differently in these two cases: in the bad news case, price wars are triggered when the signal arrives, while in the good news case, they are triggered when the signal does not arrive. It is very di;cult to provide incentives in the good news case even for a;xed small D. In fact, for small D, the probability of type I error required to provide incentives is so high that even as r S 0, no collusion is possible in AMP (AMP Proposition 4). In contrast, in our model, for any given D, as r S 0, the best equilibrium converges to;rst-best collusion. The reason is that, with a smaller r, harsher punishments allow players to use more e;cient statistical tests with cutoffs further in the tails of the price distribution. On the other hand, in the bad news case, in- centives can be provided much more e;ciently …